New Computer Fund

Thursday, July 9, 2015

Pi or not Pi

Ideal models are references and depending on just how much accuracy you need can be excellent or crap.  If you are filling a dive tank, the ideal gas laws are fantastic.  You are repeating a process done millions of times and your are generally looking at resolution of say 100 PSI.  Since the pressure on the gauge is about 14 PSI, it is negligible.  The temperature range between the tank fill site and the tank use site rarely cause more than a handful of PSI change, so there really isn't much to worry about, as long as you stay in the normal range of expectations.  When you get near the limits of the "ideal" range of the "ideal" gas laws, then you can start seeing issues provided you have gauges and such with enough precision.  You could put a high precision gauge on a dive tank and see all the fluctuations, but there really isn't any need for that precision.

This post is prompted by a new paper (PhD thesis) on CO2 forcing in the Antarctic.  That paper finds that more CO2 can cause the Antarctic to be a greater heat sink, i.e. increase the cooling rate.  This isn't anything new to me, but it is nice to see someone actually mention it in the "peer" reviewed literature.  Once you get to the extremes, about -70C and below in this cause, you started getting "other" factors come into play.  The "ideal" curve starts turning in new directions.

Believe it or not, this isn't going to have a huge impact on the "all things remaining equal" CO2 role in the Greenhouse Effect.  It will have an effect on the boring other things.

One of the other things is TSI/4 approximation.  Total Solar Irradiance (TSI) divided by 4 is a quick a dirty estimate of the energy available at at sphere for a distant point source like the sun.  It assumes you have a perfect sphere with no atmosphere or ocean to blur the lines of the spherical surface.  If Earth were a perfect sphere 1361/4 would be the "average" energy applied so Earth would have a radiant energy of about 340 Wm-2 and a temperature of about 4C degrees.  You can also use TSI/pi as an "average" energy.  That would give you an average energy of about 433Wm-2 and you can use TSI/2pi or 216 Wm-2 for another reference allowing for dark side cooling.  None of these are prefect, but they are useful references.

The TSI/4 estimate can be modified by correcting for albedo or reflection using about 0.30 or 30% reflection of solar for Earth.  To correct for reflection or albedo with TSI/pi you really should consider actual changes at the surface and not lump the whole planet together.  An example of why would be cloud cover.  Since clouds respond to surface energy, cloud cover tends to follow peak insolation.  TSI/pi already allows for incidence angle, so early morning and late evening clouds don't have much impact on the TSI/pi estimate.  Albedo close the the poles also doesn't have much impact on TSI/pi, because with angles greater than about 60 degrees there is little energy included anyway.

If you adjusted TSI/pi to allow for atmospheric refraction and TSI/4 to allow for atmospheric absorption, they would agree at a theoretical "effective" radiant layer (ERL).  So there should be an ERL of between 216 Wm-2 (day/night consideration) and 240 Wm-2 that is our "all things remaining equal" reference in the sky.  The TSI/4 fans think they have the "ideal" reference and the engineering crew think there is a range.  I am in the Engineering crew.

The "Effective" in the Effective Radiant Layer tells the engineering crew that there is sub and supra "surface" absorption/radiation and other heat transfer that can impact the accuracy of the estimates.  Sub surface would be mainly energy absorbed by the oceans at some depth that has a different residence time than energy absorbed in the atmosphere.  Supra surface would be the higher parts of the atmosphere above the theoretical ELR that have a strong radiant impact by changing the residence time of the energy in the lower atmosphere.

This is the point where theory and reality start to bite.  Both the Theorist and Engineer know that there are issues, but the two camps have difference ways for dealing with those issues.  The engineer should know there is just as real a possibility that he is wrong on the high side as the low side while the theorist tends to ignore the low side.  The Antarctic cooling in response to additional CO2 is an example of an ignored low side error. Theoretical estimates tend to be skewed towards "ideal" response, high side while engineers generally try to hedge their bets to a mid range or more Gaussian error potential.  For example, TSI/4 has a +0 and -7.5% normal error included in the Stefan-Boltzmann law.  Ideal is a limit not a range.  So TSI/4 isn't really an average it is a ideal maximum.  TSI/pi is a realistic average. Nothing wrong with using either or both as long as you "know their limitations."

I noticed with TSI/pi and the sub-surface issue, that you could model the situation as a Half-Wave rectified input with a residual DC offset.  Energy absorbed in the oceans i.e. the subsurface would have a longer residence time allowing energy accumulation not considered in a atmospheric centric model.  Basically you have a greenhouse gas effect and a greenhouse liquid effect.  You can only guess at what the residual energy stored in the oceans would be sans atmosphere, but thanks to the unique maximum density of water at 4C degree, you know that 4C is a good reference.  4C has an S-B equivalent energy of about 334 Wm-2 which would be a "DC" offset of about 118 Wm-2.

That 118 Wm-2 is just another reference and would be due to a combination of sub-surface absorption and atmospheric insolation.  BTW, none of these references consider latent heat transfer, the primary method of cooling the ocean surface or convection which is enhanced by the latent energy impact on density.  Current estimates of latent and convective energy transfer are in the ballpark of 88 Wm-2 and 25 Wm-2 respectively or about 113 Wm-2, roughly equal to the "DC" offset.  That doesn't "prove" anything but does indicate that the DC offset approach has some merit as a reference.

The combination of latent and convective heat flux driven in large part by the ocean subsurface heat retention creates a lens of sorts that blurs what has to be assumed as an "ideal" spherical surface for either the TSI/4 or TSI/pi approximations.  Which approximation best allows for that less than ideal situation should be the better reference choice.  Since TSI/pi represents a "subsurface" energy and the DC offset basically allows for the latent/convective factor, I believe more engineers would appreciate that approach.

Neither approach is perfect, but TSI/pi appears to allow for more issues.  Just my two cents.


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